Theorems

Pursuing Stacks is an influential manuscript written by Alexander Grothendieck in 1983, and of which copies were sent to Ronnie Brown and Larry Breen. It is written in English “in response to a correspondence in English”. He intended that later volumes would be in French, entitled À la poursuite des champs (literally In Pursuit of Stacks). Grothendieck gave permission to Ronnie Brown to copy the correspondence, and so this volume gradually circulated.

Preliminary to the circulated Pursuing Stacks is a 12 page letter to Daniel Quillen, dated 19-02-1983, who did not respond. The work then proceeds as a sort of research diary of about 600 pages, including many back-trackings and corrections. The ideas presented have proved hugely influential even to this day in homotopy theory, higher category theory and higher geometry, and indeed many of them have now been thoroughly worked over, ‘pursued’ and published (see below). We note however that this does not mean “Pursuing stacks” did not contain many rigorously worked notions and results itself. On the other hand one of the values of the document is to show how Grothendieck goes about developing his ideas, and he was insistent that the document should be published, if at all, “as is”, so that young people could see that even well known people could make errors.

Contents

Download

For an account of the origins of the manuscript, see Ronnie Brown’s account, where a large downloadable .ps file may be found. Scanned copies of the original typescript are available here (in djvu, warning: 23MB), here (djvu and pdf - 252MB), or here (png images) (some of these links seem no longer to work!).

Content and influence on current research

Grothendieck considered among other things, the notion of nn-groupoids and ∞\infty-groupoids, homotopy types and how to model them, homology and cohomology theories defined on categories of models and schematisation of homotopy types. This last is an attempt to define homotopy theory relative to a base ring, say, such that over ℤ\mathbb{Z} ordinary homotopy theory is recovered.

History

In 1989 Aurelio proposed to accompany me with his Alfa Romeo to visit Alexander Grothendieck in his stone hut in the middle of a lavender field near Mormoiron in the South of France. I had already visited Grothendieck in 1981 at that stone hut, but this time there was a specific item that needed to be negotiated. Grothendieck had stated that I might be the appropriate person to edit and publish his great work ‘Pursuing Stacks’. But I had very serious questions about how that editing should be done in particular. Aurelio was attentively studying the situation. The discussion was complicated by the fact that Grothendieck was bound by a religious vow of silence, so that he could only communicate by writing (though he had given me a very friendly one-­word greeting: ‘Bill !’) He wrote immediately that he was under a vow not to discuss mathematics. But his mathematical soul soon triumphed and he was writing mathematical statements and questions. (Aurelio observed all this with valiant equanimity.) The 1983 work of Grothendieck under discussion concerned among many other questions the homotopy theory of presheaf toposes. It is a long manuscript, produced by the basic method of typing all night for several weeks. Naturally, mistakes occurred due to fatigue, but the fever to press on had meant that each morning when Alexander noted the previous evening’s errors, he did not delete them, but simply kept them, and added the corrected version before proceeding. I said that as a conscientious editor I would have to delete the erroneous passages and give substantial explanations of my own. Grothendieck insisted that the errors be kept, so that students would have the opportunity to learn that even famous mathematicians make mistakes. I objected to this, pointing out that students have ample opportunity to see THAT and that learning the actual scientific material is difficult enough without the extra punishment that he was in effect advocating. But our parting was very amicable with the agreement to further consider the matter. Aurelio agreed with my point. In the end the publication did not take place. Only several years later did Maltsiniotis and Cisinski work out the mathematics and bring this magnificent homotopy program to an initial fruition.